Linear quantitative CFI conjecture for $C_q$ weights

Let $1<p<q$, let $w\in C_q$, let $T$ be a Calderón--Zygmund operator, and let $f\in C_c^\infty(\mathbb{R}^n)$. Linear quantitative CFI conjecture. The estimate

$$\left\lVert T^*f\right\rVert_{L^p(w)}\leq c_{n,T,p,q} ( [w]_{C_q}+1) \left\lVert M f\right\rVert_{L^p(w)}$$

should hold. The paper's preceding theorem proves the same estimate with an additional factor $\log(e+[w]_{C_q})$; the conjecture asks for linear dependence on the $C_q$ characteristic.